# Why is an algebra over a ring called an algebra? [duplicate]

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After having read this post on MathOverflow about the etymology of various terms that are used in abstract algebra like groups, rings and fields, I was wondering how and why did the term algebra come to denote specifically algebra over a field or a commutative ring? Why pick the same name as the discipline to denote a very specific algebraic structure?

This question might not be very useful mathematically, but I'd nevertheless like to know the origin of this nomenclature.

## marked as duplicate by Simon S, Mnifldz, Community♦Aug 10 '15 at 16:03

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• With the introduction of History of Science & Mathematics Stack Exchange I've often wondered if questions like this should end up there. – user83387 Aug 10 '15 at 14:38
• @shuckles I did not know about the existence of History of Science and Mathematics SE. Thanks for the info..:) – sayantankhan Aug 10 '15 at 16:03

## 1 Answer

A. A. Albert introduced a $K$-algebra $A$ in the sense that $A$ is a $K$-vector space with distributive bilinear product $A\times A\rightarrow A$, $(a,b)\mapsto a\cdot b$. This includes Lie algebras, Jordan algebras, associative algebras, commutative algebras etc. A natural generalization is to replace the field $K$ by a commutative ring $R$ with $1$. For example, one can consider Lie algebras over the ring $\mathbb{Z}$ (although the name Lie ring is used, too).
The same applies to groups, say $GL_n(K)$ over a field $K$. It can be generalized to $GL_n(R)$ over a commutative ring $R$, and still is called a group ("why is a group over a ring called a group - and is not a group ring").